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MULTIPLE TRANSFORMATIONS In part (b) of Example 2, graphing y 523⏐x⏐
involves both vertically stretching and reflecting the graph of y 5 ⏐x⏐. A graph
may be related to a parent graph by even more than two transformations. For
example, the graph of y 5 a⏐x 2 h⏐ 1 k can involve a vertical stretch or shrink, a
reflection, and a translation of the graph of y 5 ⏐x⏐.
E XAMPLE 3 Graph a function of the form y 5 a⏐x 2 h⏐ 1 k
Graph y 522⏐x 2 1⏐ 1 3. Compare the graph with the graph of y 5 ⏐x⏐.
Solution
STEP 1 Identify and plot the vertex, (h, k) 5 (1, 3).
STEP 2 Plot another point on the graph, such as
(0, 1). Use symmetry to plot a third point,
(2, 1).
STEP 3 Connect the points with a V-shaped graph.
STEP 4 Compare with y 5 ⏐x⏐. The graph of
y 522⏐x 2 1⏐ 1 3 is the graph of y 5 ⏐x⏐
stretched vertically by a factor of 2, then
y 52 2z x 2 1 z 1 3
reflected in the x-axis, and finally translated right 1 unit and up 3 units.
✓ GUIDED PRACTICE for Examples 1, 2, and 3
Graph the function. Compare the graph with the graph of y 5 ⏐x⏐.
1. y 5 ⏐x 2 2⏐ 1 5 2. y 5 1 } 4 ⏐x⏐ 3. f(x) 523⏐x 1 1⏐ 2 2
E XAMPLE 4 Write an absolute value function
HOLOGRAMS In holography, light
from a laser beam is split into two
beams, a reference beam and an
object beam. Light from the object
beam reflects off an object and is
recombined with the reference beam
to form images on film that can be
used to create three-dimensional
images. Write an equation for the
path of the reference beam.
Solution
The vertex of the path of the reference
beam is (5, 8). So, the equation has the
form y 5 a⏐x 2 5⏐ 1 8. Substitute the
coordinates of the point (0, 0) into the equation and solve for a.
0 5 a⏐0 2 5⏐ 1 8 Substitute 0 for y and 0 for x.
21.6 5 a Solve for a.
c An equation for the path of the reference beam is y 521.6⏐x 2 5⏐ 1 8.
y
(1, 3)
y 5 z x z
2.7 Use Absolute Value Functions and Transformations 125
2
1
x